A characterization of the symmetric steady water waves in terms of the underlying flow

In this paper we present a characterization of the symmetric rotational periodic gravity water
waves of finite depth and without stagnation points in terms of the underlying flow.
Namely, we show that such a wave is symmetric and has a single crest and trough per period if and only if
there exists a vertical line within the fluid domain such that all
the fluid particles located on that line minimize there simultaneously their distance to the fluid bed
as they move about.
Our analysis uses the moving plane method, sharp elliptic maximum principles, and the principle of analytic continuation.